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UNIFIED ACCELERATION OF high-order algorithms UNDER GENERAL HO"\LDER CONTINUITY

SIAM JOURNAL ON OPTIMIZATION 2021年第3期31卷 1797-1826页

作者： Song, Chaobing Jiang, Yong Ma, Yi Tsinghua Univ Tsinghua Berkeley Shenzhen Inst Shenzhen 518055 Guangdong Peoples R China Univ Calif Berkeley Dept Elect Engn & Comp Sci Berkeley CA 94720 USA

In this paper, through an intuitive vanilla proximal method perspective, we derive a concise unified acceleration framework (UAF) for minimizing a convex function that has Ho"\lder continuous derivatives with respect to general (non-Euclidean) norms. The UAF reconciles two different high-order acceleration approaches, one by Nesterov [Math. Program., 112 (2008), pp. 159181] and one by Monteiro and Svaiter [SIAM J. Optim., 23 (2013), pp. 1092-1125]. As a result, the UAF unifies the high-order acceleration instances of the two approaches by only two problem related parameters and two additional parameters for framework design. Meanwhile, the UAF (and its analysis) is the first approach to make high-order methods applicable for high-order smoothness conditions with respect to non-Euclidean norms. Furthermore, the UAF is the first approach that can match the existing lower bound of iteration complexity for minimizing a convex function with Ho"\lder continuous derivatives. For practical implementation, we introduce a new and effective heuristic that significantly simplifies the binary search procedure required by the framework. We use experiments on logistic regression to verify the effectiveness of the heuristic. Finally, the UAF is proposed directly in the general composite convex setting and shows that the existing high-order algorithms can be naturally extended to the general composite convex setting.

关键词： Key words high-order algorithms Nesterov's acceleration proximal method non-Euclidean norm

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high-order algorithms FOR RIESZ DERIVATIVE AND THEIR APPLICATIONS (III)

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FRACTIONAL CALCULUS AND APPLIED ANALYSIS 2016年第1期19卷 19-55页

作者： Ding, Hengfei Li, Changpin Tianshui Normal Univ Sch Math & Stat Tianshui 741001 Peoples R China Shanghai Univ Dept Math Shanghai 200444 Peoples R China

Numerical methods for fractional calculus attract increasing interest due to its wide applications in various fields such as physics, mechanics, etc. In this paper, we focus on constructing high-order algorithms for Riesz derivatives, where the convergence orders cover from the second order to the sixth order. Then we apply the established schemes to the Riesz type turbulent diffusion equation (or, Riesz space fractional turbulent diffusion equation). Numerical experiments are displayed which support the theoretical analysis.

关键词： space fractional turbulent diffusion equation high-order algorithms stability analysis

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Combined high-order algorithms in robust least-squares estimation with harmonic regressor and strictly diagonally dominant information matrix

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PROCEEDINGS OF THE INSTITUTION OF MECHANICAL ENGINEERS PART I-JOURNAL OF SYSTEMS AND CONTROL ENGINEERING 2015年第2期229卷 184-190页

作者： Stotsky, Alexander Chalmers Environm & Energy Dept Div Elect Power Engn SE-41296 Gothenburg Sweden

This article describes new high-order algorithms in the least-squares problem with harmonic regressor and strictly diagonally dominant information matrix. Estimation accuracy and the number of steps to achieve this accuracy are controllable in these algorithms. Simplified forms of the high-order matrix inversion algorithms and the high-order algorithms of direct calculation of the parameter vector are found. The algorithms are presented as recursive procedures driven by estimation errors multiplied by the gain matrices, which can be seen as preconditioners. A simple and recursive (with respect to order) algorithm for update of the gain matrix, which is associated with Neumann series, is found. It is shown that the limiting form of the algorithm (algorithm of infinite order) provides perfect estimation. A new form of the gain matrix is also a basis for unification method of high-order algorithms. New combined and fast convergent high-order algorithms of recursive matrix inversion and algorithms of direct calculation of the parameter vector are presented. The stability of algorithms is proved and explicit transient bound on estimation error is calculated. New algorithms are simple, fast and robust with respect to round-off error accumulation.

关键词： Least-squares estimation oscillating signals harmonic regressor strictly diagonally dominant matrix high-order algorithms recursive matrix inversion preconditioning Neumann series strictly diagonally dominant solver

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Analysis of inhomogeneous structures in small and large deformations using the finite element-meshless coupling method

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COMPUTERS & MATHEMATICS WITH APPLICATIONS 2024年 169卷 273-297页

作者： El Kadmiri, Redouane Belaasilia, Youssef Timesli, Abdelaziz Hassan II Univ Casablanca Natl Higher Sch Arts & Crafts ENSAM CASABLANCA Casablanca 20670 Morocco Chouaib Doukkali Univ Fac Sci Lab Nucl Atom Mol Mech & Energet Phys El Jadida Morocco

In this work, a finite element-meshless coupling method for modeling the inhomogeneous structures composed of functionally graded materials is presented. Coupling the two methods is usually based on the continuity and equilibrium conditions at the finite element-meshless interface. In the proposed hybrid method, the equilibrium condition is satisfied by the action-reaction principle to ensure the coupling between the finite element method and the strong-form meshless method. The idea is to satisfy the equilibrium condition by calculating the interface force vector in the finite element formulation using a strong-form meshless technique. In the weak formulation the forces are modeled by the interface force functional. This hybrid method has been used to study the behavior of functionally graded materials in small deformations, with a comparison between the results of other numerical methods and those of analytical solutions. Following this validation, the hybrid numerical approach is adapted to the simulation of static computational problems for inhomogeneous functionally graded materials with geometric nonlinearity. In this context, a high-order algorithm has been developed by associating a high-order development, continuation procedure and hybrid method. Numerical analysis is performed to prove the accuracy and efficiency of the present approach, through a comparative study with solutions provided by high-order algorithms derived from the finite element method and strong-form meshless methods. In addition, the good qualities of the solution are controlled by the residues, which do not exceed 10-6. -6 .

关键词： Mixed finite element/meshless method Finite element method Strong form meshless methods high-order algorithms Inhomogeneous structures Functionally graded materials

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CONTRACTING PROXIMAL METHODS FOR SMOOTH CONVEX OPTIMIZATION

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SIAM JOURNAL ON OPTIMIZATION 2020年第4期30卷 3146-3169页

作者： Doikov, Nikita Nesterov, Yurii Catholic Univ Louvain UCL Inst Informat & Commun Technol Elect & Appl Math B-1348 Louvain La Neuve Belgium Catholic Univ Louvain UCL Ctr Operat Res & Econometr CORE B-1348 Louvain La Neuve Belgium

In this paper, we propose new accelerated methods for smooth convex optimization, called contracting proximal methods. At every step of these methods, we need to minimize a contracted version of the objective function augmented by a regularization term in the form of Bregman divergence. We provide global convergence analysis for a general scheme admitting inexactness in solving the auxiliary subproblem. In the case of using for this purpose high-order tensor methods, we demonstrate an acceleration effect for both convex and uniformly convex composite objective functions. Thus, our construction explains acceleration for methods of any order starting from one. The augmentation of the number of calls of oracle due to computing the contracted proximal steps is limited by the logarithmic factor in the worst-case complexity bound.

关键词： convex optimization proximal method accelerated methods global complexity bounds high-order algorithms

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high-order Interpolation algorithms for Charge Conservation in Particle-in-Cell Simulations

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Communications in Computational Physics 2013年第4期13卷 1134-1150页

作者： Jinqing Yu Xiaolin Jin Weimin Zhou Bin Li Yuqiu Gu Vacuum Electronics National Laboratory University of Electronic Science and Technology of ChinaChengdu 610054China Research Center of Laser Fusion China Academy of Engineering PhysicsMianyang 621900China

high-order interpolation algorithms for charge conservation in Particle-inCell(PIC)simulations are *** methods are valid for the case that a particle trajectory is a zigzag *** second-order and third-order algorithms which can be applied to any even-order and odd-order are discussed in this paper,*** test simulations are performed to demonstrate their validity in two-dimensional PIC *** with the simulation results of one-order,high-order algorithms have advantages in computation precision and enlarging the grid scales which reduces the CPU time.

关键词： high-order algorithms charge conservation PIC code CPU time

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Towards accurate estimation of fast varying frequency in future electricity networks: The transition from model-free methods to model-based approach

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PROCEEDINGS OF THE INSTITUTION OF MECHANICAL ENGINEERS PART I-JOURNAL OF SYSTEMS AND CONTROL ENGINEERING 2016年第10期230卷 1164-1175页

作者： Stotsky, Alexander Chalmers Div Elect Power Engn Environm & Energy Dept SE-41296 Gothenburg Sweden

Accurate estimation of fast varying fundamental frequency in the presence of harmonics and noise will be required for effective frequency regulation in future electricity networks with high penetration level of renewable energy sources. Two new algorithms for network frequency tracking are proposed. The first algorithm represents a robust modification of classical zero crossing method, which is widely used in industry. The second algorithm is a multiple model algorithm based on the systems with harmonic regressor. Algorithm allows complete reconstruction of the frequency content of the signal, using information about the upper bound of the number of harmonics only. Moreover, new family of high-order algorithms together with new stepwise splitting method are proposed for parameter calculation in systems with harmonic regressor for the accuracy improvement. Statistical methods are introduced for comparison of two new algorithms to classical zero crossing algorithm. The modified algorithm provides significant improvement compared to the classical algorithm, and the algorithm with harmonic regressor provides further improvement of the statistical performance indexes with respect to the modified algorithm.

关键词： Accurate frequency tracking in future electricity networks harmonic regressor multiple model statistical quantification methods of comparison of algorithms high-order algorithms stepwise splitting inversion of ill-conditioned matrices

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A high-order multi-variable Fuzzy Time Series forecasting algorithm based on fuzzy clustering

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EXPERT SYSTEMS WITH APPLICATIONS 2015年第4期42卷 2121-2135页

作者： Askari, S. Montazerin, N. Amirkabir Univ Technol Dept Mech Engn Tehran Polytech Tehran *** Iran

A high-order algorithm for Multi-Variable Fuzzy Time Series (HMV-FTS) is presented based on fuzzy clustering to eliminate some well-known problems with the existing FTS algorithms. high-order algorithms can handle only one-variable FTS and multi-variable algorithms can handle only one-order FITS. HMV-FTS does both tasks simultaneously. FITS algorithms cannot incorporate existing information about future value of a variable in the forecasting process while HMV-FTS can. Defuzzification of the fuzzy value of a forecast to cluster centers or midpoint of intervals and use of intervals are other controversial problems with the existing FTS algorithms. These are eliminated by constructing fuzzy sets from partition matrices and letting each data point to contribute in defuzzification based on its membership grade in the fuzzy sets. In multi-variable FTS algorithms, one variable is considered as main variable which is forecasted and the other variables are secondary;while HMV-FTS treats all variables equally and more than one variable can be forecasted at the same time. It is shown that HMV-FTS is suitable for system identification, forecasting and interpolation. This algorithm is more accurate than popular FTS algorithms and other forecasting tools and systems such as ANFIS, Type II fuzzy model and ARIMA model. (C) 2014 Elsevier Ltd. All rights reserved.

关键词： Fuzzy sets Fuzzy Time Series Fuzzy clustering Fuzzy C-Means Multi-variable algorithms high-order algorithms

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GPU-Accelerated Efficient Implementation of FDTD Methods With Optimum Time-Step Selection

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IEEE TRANSACTIONS ON MAGNETICS 2014年第2期50卷 477-480页

作者： Zygiridis, Theodoros T. Kantartzis, Nikolaos V. Tsiboukis, Theodoros D. Univ Western Macedonia Dept Informat & Telecommun Engn Karamanli 50100 Kozani Greece Univ Western Macedonia Dept Informat & Telecommun Engn Lygeris 50100 Kozani Greece Aristotle Univ Thessaloniki Dept Elect & Comp Engn Thessaloniki 54124 Greece

The consistent combination of uneven space-time orders in finite-difference time-domain (FDTD) algorithms is the subject of this paper. When low-order time integration is used in conjunction with high-order spatial expressions, the operation of the numerical scheme close to the stability limit causes degraded performance and slow convergence. By exploiting accuracy considerations, we derive an estimate of the optimum-much smaller-time-step size that ameliorates errors in a mean-value sense and leads to improved precision. To deal with the augmentation of the required iterations, the parallel implementation of the FDTD techniques on graphics processing units is pursued, ensuring faster code executions and more efficient models.

关键词： FDTD method GPU computing high-order algorithms time stepping

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Front Tracking Method with high-order Enhancement and Its Application in Two-Phase Micromixing of Incompressible Viscous Fluids

Front Tracking Method with High-Order Enhancement and Its Ap...

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作者： Yijie Zhou Stony Brook University

学位级别：博士

We develop a front tracking method based on the hydrodynamic library FronTier for the solution of the governing equations of motion for two-phase micromixing of incompressible, viscous, liquid-liquid solvent extraction pro- cesses. The method is used for accurate simulation of the turbulent micromix- ing dynamics of an aqueous and an organic phase exposed to intense centrifugal force and shearing stress. The onset of mixing is the result of the combina- tion of the classical Rayleigh-Taylor and Kelvin-Helmholtz instabilities. We demonstrate verification and convergence results for one-phase and unmixed, two-phase flows. For mixed, two-phase flow a mixing environment that em- ulates a sector of the annular mixing zone of a centrifugal contactor is used with the mathematical domain small enough to allow for resolution of the indi- vidual interfacial structures and large enough to allow for an analysis of their statistical distribution of sizes and shapes. Such a statistical picture provides the information needed for building a consistent coarsened model applicable to the entire mixing device. We reach a stable two phase configuration as a sta- tistically steady state in late time after going through a fully mixed transient chaotic flow regime with a high surface area. To handle problems introduced by the extreme complexity of interfaces, a new parallel triangular mesh li- brary called HiProp is implemented which serves as the basis for high-order mesh algorithms. The new library keeps a full list of parallel information for each point and triangle so that each element has a unique master processor and global ID. No floating point comparison is needed after the parallel in- formation is built. The utilities for building ghost triangles while keeping the parallel information updated based on either connectivity or domain decompo- sition are implemented for applying different high-order mesh algorithms. We develop parallel high-order mesh smoothing, parallel high-order

关键词： front tracking Taylor-Couette flow turbulent fluid flow mixing parallel triangular mesh high-order algorithms

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